Discrete analytic functions: convergence theorems

Complex analysis on an arbitrary graph lying in the complex plane and having convex quadrilateral faces is developed. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal. This generalizes the definition by R. Isaacs, R. Duffin, and C. Mercat.
We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution converges to a harmonic function in the scaling limit (under certain regularity assumptions). This solves a problem of S. Smirnov from 2010.
This was proved earlier by R. Courant–K. Friedrichs–H. Lewy for square lattices and by D. Chelkak–S. Smirnov for rhombic lattices. The result provides a new approximation algorithm for numerical solution of the Dirichlet boundary value problem and presumably some probabilistic corollaries. The proof is based on energy estimates inspired by alternating-current networks theory.